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Finding the optimal design of experiments in the Bayesian setting typicallyrequires estimation and optimization of the expected information gainfunctional. This functional consists of one outer and one inner integral,separated by the logarithm function applied to the inner integral. When themathematical model of the experiment contains uncertainty about the parametersof interest and nuisance uncertainty, (i.e., uncertainty about parameters thataffect the model but are not themselves of interest to the experimenter), twoinner integrals must be estimated. Thus, the already considerable computationaleffort required to determine good approximations of the expected informationgain is increased further. The Laplace approximation has been appliedsuccessfully in the context of experimental design in various ways, and wepropose two novel estimators featuring the Laplace approximation to alleviatethe computational burden of both inner integrals considerably. The firstestimator applies Laplace's method followed by a Laplace approximation,introducing a bias. The second estimator uses two Laplace approximations asimportance sampling measures for Monte Carlo approximations of the innerintegrals. Both estimators use Monte Carlo approximation for the remainingouter integral estimation. We provide three numerical examples demonstratingthe applicability and effectiveness of our proposed estimators.

Laplace-based strategies for Bayesian optimal experimental design with nuisance uncertainty